Saying "port port" is clumsy. Should I refer to it as the port berthing mechanism? Is there specific terminology for what I'm referring to or are there a few options? Also, I'm specifically referencing modules berthed to each other in this case, not supply ships docking to modules. Is there a difference? (FYI, port:left :: starboard:right)
They would most likely use a TLA (Three Letter Acronym) instead of the noun "port". In the example you give, they would say the "port CBM" (Common Berthing Mechanism).
Here's an example from the STS-130 Flight Plan